🌿 Sagebrush Atlas
Atlas β€Ί Modular forms β€Ί Weight 2 β€Ί Level 839 β€Ί 839.2.a β€Ί 839.2.a.a

Newform orbit 839.2.a.a

Trace form

The 19 newforms in this orbit are conjugate over Q\mathbb{Q}; the sum of their qq-expansions is

19qβˆ’3q2βˆ’4q3+7q4βˆ’4q5βˆ’9q6βˆ’13q7βˆ’6q8βˆ’5q9βˆ’10q10βˆ’12q11βˆ’4q12βˆ’16q13βˆ’2q14βˆ’7q15βˆ’13q16βˆ’13q17βˆ’3q18βˆ’20q19βˆ’8q20βˆ’19q21βˆ’12q22βˆ’17q24+O(q25)19q - 3q^{2} - 4q^{3} + 7q^{4} - 4q^{5} - 9q^{6} - 13q^{7} - 6q^{8} - 5q^{9} - 10q^{10} - 12q^{11} - 4q^{12} - 16q^{13} - 2q^{14} - 7q^{15} - 13q^{16} - 13q^{17} - 3q^{18} - 20q^{19} - 8q^{20} - 19q^{21} - 12q^{22} - 17q^{24} + O(q^{25})

Coefficient field

The coefficients ana_n generate a number field of degree 19. It is generated by the eigenvalue Ξ±\alpha of T=T2+T3T = T_{2} + T_{3}, whose minimal polynomial is

x19+7x18βˆ’15x17βˆ’179x16+x15+1728x14+845x13βˆ’8045x12βˆ’3834x11+19493x10+4887x9βˆ’23331x8βˆ’797x7+12253x6βˆ’1037x5βˆ’2314x4+202x3+140x2βˆ’2xβˆ’1x^{19} + 7x^{18} - 15x^{17} - 179x^{16} + x^{15} + 1728x^{14} + 845x^{13} - 8045x^{12} - 3834x^{11} + 19493x^{10} + 4887x^{9} - 23331x^{8} - 797x^{7} + 12253x^{6} - 1037x^{5} - 2314x^{4} + 202x^{3} + 140x^{2} - 2x - 1

Atkin–Lehner signs

ppwpw_p
839+1+1

For pβˆ₯Np \parallel N every form in the orbit has ap=βˆ’pk/2βˆ’1wpa_p = -p^{k/2-1} w_p. The Fricke sign is wN=+1w_N = +1, so the sign of the functional equation is (βˆ’1)k/2wN=βˆ’1(-1)^{k/2} w_N = -1: each L-function in the orbit has odd analytic rank.

Traces of Hecke eigenvalues

pp2357111317192329313741434753596167717379838997
tr⁑ap\operatorname{tr} a_pβˆ’3βˆ’4βˆ’4βˆ’13βˆ’12βˆ’16βˆ’13βˆ’200βˆ’20βˆ’21βˆ’32βˆ’39βˆ’207βˆ’711βˆ’40βˆ’9βˆ’14βˆ’56βˆ’457βˆ’9βˆ’42

tr⁑an\operatorname{tr} a_n for every n≀1000n \le 1000 is in the JSON.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.274 s (one thread) and proven: the dimensions are certified, and the characteristic polynomial of TT over Z\mathbb{Z} is reconstructed by CRT past a rigorous coefficient bound. The label, dimension and tr⁑ap\operatorname{tr} a_p for every p<1000p < 1000 equal LMFDB's.

The engine's checks
  • dim M^+ at levels [1, 839]: [0, 71] (certified at ell = 2147479751)
  • T = [(2, 1), (3, 1)]; h of degree 70 = 1 x 70 (phi(ord eps) x dim S_k^new); 7 primes, 0 rejected
Open in the notebook Β· JSON Β· stored data
Recompute it yourself

In Sage mode at sagebrush.space (or npx sagebrush --sage):

newform_orbits(839, 2)[0]   # 839.2.a.a

With the Rust engine (source), in engine/:

cargo run --release -p sagebrush-web --example atlas -- 839:2 1000